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I was helping my niece (7th grade) with homework and one of the topics was the absolute mean deviation. It's basically the same thing as standard deviation except instead of squaring the difference between the data points and the mean, you take the absolute value of them.

On the one hand I can almost understand this because if I told her she had to square all those numbers instead of just dropping negative signs, she wouldn't have liked it. That being said, she was using a calculator so squaring the numbers and dealing with big numbers should have been trivial anyway. The absolute mean deviation isn't something that is used by even statisticians (I don't think) but standard deviation is pretty common in general business (at the very least) so why teach 90% of a real world topic instead of just teaching the real thing?

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    $\begingroup$ Maybe they will get to SD next. You don't have to take it as excluding A to be shown B. There are really a lot of ways to look at a distribution (mean, median, range, SD, max, min, upper or lower constraints, etc.) Also, for some problems absolute deviation from a standard is exactly what you want to know, not SD (e.g. for physical or $ problems, at times.) $\endgroup$ – guest Jun 8 '18 at 11:36
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    $\begingroup$ Absolute error is definitely used by statisticians. $\endgroup$ – Daniel R. Collins Jun 8 '18 at 13:40
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    $\begingroup$ I wouldn't say this to most students (which is why I'm putting it in a comment rather than in an answer) but it only recently occurred to me that the relationship between "standard deviation" and "mean absolute deviation" is almost precisely the same as the relationship between "Euclidean geometry" and "taxicab geometry". $\endgroup$ – mweiss Jun 8 '18 at 23:26
  • $\begingroup$ @guest they didn't $\endgroup$ – Dean MacGregor Jun 8 '18 at 23:31
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    $\begingroup$ They didn't THAT YEAR. Later means later. We are talking about education of a child. $\endgroup$ – guest Jun 9 '18 at 3:50
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The question we pose to students is: How far away, on average, are these values from their mean?

The "natural" way to answer that question is to compute the deviations of the individual data points and compute the average. When students do this they are usually surprised to find out that the answer comes out to be precisely $0$ — that the positive deviations and the negative deviations always cancel exactly. Pointing out that this is actually a signal that we did everything right -- that the mean value is supposed to be in the middle of the data, with just as much deviation above it as below it -- often triggers an "aha" moment for many students, for whom the average was just a formula disconnected from any intuitive meaning (see How to Teach Averages (Arithmetic Mean) to a Teenager? and https://math.stackexchange.com/q/922028/124095 for examples of this).

So, once we have established that "average deviation" is not a useful thing to compute, the obvious next question is: how could we get rid of those negative signs? The idea of just dropping them all is both simple and intuitive, and "average of the absolute values of the deviations" is a literal interpretation of the idea of "average distance from the mean".

In fact, as others have pointed out, it's the standard deviation that seems unnatural and unnecessarily complicated. We square the deviations, which makes them all positive, and then we average them, but that comes out too big because we squared them, so then we take the square root... Why do all of that? I usually tell my students that the function $y=x^2$ has nicer mathematical properties than the function $y=|x|$, and in particular that the sharp corner in the absolute value function makes it hard to work with. I'm not sure how intellectually honest that response is, but it at least makes a plausible-seeming connection with Calculus (for the students who have seen it) or lays some groundwork for it (for the students who have not yet but may someday).

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  • $\begingroup$ That's what I say too! In fact, when I was first using standard deviation in an applied algebra course I was teaching, it seemed wrong to me. I still don't have enough depth in statistics to understand whether it really is right, or is just a kludge. $\endgroup$ – Sue VanHattum Jun 12 '18 at 20:15
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The foundations of statistical inference are very hard to teach at any level, and almost certainly, at the 7th grade level, little or no serious motivation is given for the rules presentd. Probably at the 7th grade level what is being taught are rote rules for calculation of defined, but generally poorly if at all motivated, quantities. The presence or absence of a square is hardly of much importance if that is the situation.

In quantifying error, the power used refects some subjective judgment about the importance attached to large values or outliers. One can use the absolute value, the square, or any $p$th power (or the $\infty$-norm). Different choices are affected differently by large values - the mean error calculated using a higher power is more affected by large values than is the error calculated using a lower power. The widespread use of the square has sound justifications in scientific practice, but the ordinary mean also can make sense. When working with large data sets on a computer, computational considerations can dictate using the square. Fundamentally the preference for the square is because the squared mean is given by an inner product. For example, the regression line defined using mean squared error is more easily calculated on a computer than that defined using mean absolute error because the "normal equations" can be solved efficiently and robustly using the singular value decomposition.

On the other hand, if one refers to computation by hand, then the calculation without squares requires fewer operations and so is simpler. At the $7$th grade level little or no effort is made to interpret, and to the extent that any is made, the difference between ordinary mean and mean squared is not of great importance.

There is nothing any more "real world" about any of this stuff than there is about the standard exercises given to calculus students. One teaches mathematics to all $7$th graders not because of its real world utility, but because the habits of thought learned by learning mathematics are themselves useful quite generally. One teaches elementary statistical ideas in order to develop the ability to formulate qualitative interpretations of masses of data. What one can hope to teach students is that there is some sort of second order criterion that can distinguish distributions with the same mean. One hopes that a student can distinguish (visually/intuitively) between data uniformly distributed over a range and data all repeated on the midpoint of that same range, and then see that the measure of deviation (with or without squares) differentiates between these two data points. This is in the same spirit as teaching the student that distributions with the same mean can have different medians, but it is different, because mean and median are both first order statistics, whereas deviation is somehow second order, measuring spread around something first order. There are different ways to measure spread, but it becomes difficult to justify, on principled grounds understandable at the middle school level, the choice of one measure of spread over another. If I were teaching this material, I would probably use the mean squared error as the default, but I have little experience teaching $7$th graders per se, and am prepared to imagine the leaving out the square facilitates the realization of computations with little conceptual loss of a nature relevant at that level.

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You're in the U.S. (according to your profile), and in the U.S. square roots are not generally introduced until the 8th grade (for example, 8.EE.A.2 in the Common Core). I believe the topic "absolute mean deviation" (also called "mean absolute deviation") is a relatively recent addition to middle grade level mathematics, probably within the last 10 years, that allows teachers to give a more nuanced measure of data variability than "range" and to give a cognitive stepping stone to "standard deviation" (taught in high school), before students have been introduced to square roots. In the Common Core, absolute mean deviation is taught in the 6th and 7th grades (6.SP.B.5.c and 7.SP.B.3).

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When I introduce measures of dispersion, the usual question from students is why do we use standard deviation and variance instead of absolute deviation, which is a lot simpler to interpret and compute. That is, they ask the opposite as the OP.

Since we know more statistics (and calculus), we know that absolute deviation is not very useful. However, when introducing measures of dispersion, it's useful because it's one of the most intuitive measures and understanding it helps introducing and motivating variance and standard deviation.

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  • $\begingroup$ Another reason for favoring variance/s.d. is because we keep bumping into the normal distribution - thanks CLT! - and variance/s.d. is one of the two parameters that characterize normal distributions. $\endgroup$ – pjs Jun 10 '18 at 14:16
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    $\begingroup$ @pjs I agree, but please notice that normal distribution could be easily parametrized using absolute deviation, too. We would just need to multiply by a constant: math.stackexchange.com/questions/1850653/… $\endgroup$ – Pere Jun 10 '18 at 14:29
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Consider, when constructing Bayes estimators:

  • Using the squared error loss gives the basis for the Bayes estimator of the mean.
  • Using the absolute error loss gives the basis for the Bayes estimator of the median.

From: Casella/Berger, Statistical Inference, Chapter 10, Theorem 10.3.2.

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